Digital Signature Technique

ABSTRACT

A method for signing a digital message, including the following steps: selecting parameters that include first and second primes, a ring of polynomials related to the primes, and at least one range-defining integer; deriving private and public keys respectively related to a random polynomial private key of the ring of polynomials, and to evaluations of roots of unity of the random polynomial to obtain a public key set of integers; storing the private key and publishing the public key; signing the digital message by: (A) generating a noise polynomial, (B) deriving a candidate signature by obtaining a hash of the digital message and the public key evaluated at the noise polynomial, and determining the candidate signature using the private key, a polynomial derived from the hash, and the noise polynomial, (C) determining whether the coefficients of the candidate signature are in a predetermined range dependent on the at least one range-defining integer, and (D) repeating steps (A) through (C) until the criterion of step (C) is satisfied, and outputting the resultant candidate signature as an encoded signed message.

RELATED APPLICATION

This application claims priority from U.S. Provisional Patent Application No. 61/958,221 filed Jul. 23, 2013, and said Provisional patent application is incorporated herein by reference.

FIELD OF THE INVENTION

This invention relates to the field of cryptography and, more particularly, to a public key digital signature technique.

BACKGROUND OF THE INVENTION

Public key digital signatures are important for secure exchange of information between plural parties, for example between computers or mobile devices, or between a smart card and a terminal.

In the late 1990s two of the inventors hereof proposed authentication and signature schemes based on the problem of recovering a polynomial with tightly concentrated coefficients given a small number of evaluations of that polynomial. The heuristic justification for the security of the scheme was that the uncertainty principle severely restricts how concentrated a signal can be in two mutually incoherent bases.

An early incarnation of that scheme is described in U.S. Pat. No. 6,076,163. and a later version, called PASS-2 was described in Hoffstein, J., Silverman, J. H.: Polynomial Rings and Efficient Public Key Authentication II. In: Lam, K. Y., Shparlikski, I., Wang, H., Xing, C. (eds.), Cryptography and Computational Number Theory, Progress in Computer Science and Applied Logic, vol. 20, pp. 269-286, Birkhauser (2001). A summary description of the PASS-2 technique is included as part of the attached Appendix I. The original PASS protocols, which are also described in Appendix I, include the following: Given a message μ, a secret key f with small norm, and a public key {circumflex over (f)}l_(Ω=) f _(Ω)f, equal to the evaluations of f at the values contained in the set Ω, the objective is to construct a signature that mixes f and μ and can be verified by means of {circumflex over (f)}l_(Ω). A prototype of this was presented in the above-referenced U.S. Pat. No. 6,076,163.

To sign, the signer

-   -   Computes and keeps secret a short polynomial gεR_(q) and reveals         the commitment ĝ/_(Ω)=F_(Ω)g.     -   Computes and reveals a short challenge polynomial cεR_(q) from         Hash(ĝl_(Ω),μ).     -   Computes and reveals h=g*(f+c).

To verify, the verifier

-   -   Verifies that h has norm less than a specific upper bound.     -   Verifies that c=Hash(ĥl_(Ω)/({circumflex over         (f)}l_(Ω)+ĉl_(Ω)),μ)

The first condition for verification is met because

|g*(f+c)|≈|g∥f+c|.

The fact that |f|, |g|, |c| are small thus implies that Ihl is small. The second condition is true because

_(Ω) is a ring homomorphism.

To forge a signature, a third party would need to produce an h which is short, and which satisfies the required evaluations at points in Ω. It is conjectured that finding such an h is no easier than solving the associated closest vector problem.

The difficulty with this PASS prototype is that a transcript of signatures produced by a single signer on any set of messages leaks information about that signer's secret key. This is explained further in Appendix I.

The problem with PASS was not that individual signatures leaked information about the secret key, but rather that an average over a collection of signatures would converge to a secret key dependent value.

It is among the objects of the present invention to address and solve this type of vulnerability in certain public key digital signature techniques.

SUMMARY OF THE INVENTION

In accordance with an aspect of an embodiment of the invention, a PASS type of digital signature technique is devised which employs rejection sampling that assures that transcript distributions are completely decoupled from the keys that generate them. Background rejection sampling is described, for example, in Lyubashevsky, V., Fiat-Shamir With Aborts, Applications to Lattice and Factoring-Based Signatures, In: ASIACRYPT 2009, pp. 598-616. Springer (2009).

In accordance with an embodiment of the invention, a method is set forth for signing and subsequently verifying a digital message, including the following steps implemented using at least one processor-based subsystem: selecting parameters that include first and second primes, a ring of polynomials related to said primes, and at least one range-defining integer; deriving private and public keys respectively related to a random polynomial private key of the ring of polynomials, and to evaluations of roots of unity of the random polynomial to obtain a public key set of integers; storing the private key and publishing the public key; signing the digital message by: (A) generating a noise polynomial, (B) deriving a candidate signature by obtaining a hash of the digital message and the public key evaluated at the noise polynomial, and determining the candidate signature using the private key, a polynomial derived from the hash, and the noise polynomial, (C) determining whether the coefficients of the candidate signature are in a predetermined range dependent on said at least one range-defining integer, and (D) repeating steps (A) through (C) until the criterion of step (C) is satisfied, and outputting the resultant candidate signature as an encoded signed message; and performing a verification procedure utilizing the encoded signed message and the public key to determine whether the encoded signed message is valid.

In a disclosed embodiment of the invention, said step of selecting parameters that include at least one range-defining integer comprises selecting parameters that include first and second range-defining integers, and the step (C) of said signing of the digital message comprises determining whether the coefficients of the candidate signature are in a predetermined range dependent on said first and second range-defining integers. In this embodiment, the first and second range-defining integers define norm bound ranges, and the step of determining whether the coefficients of the candidate signature are in a predetermined range comprises determining whether said coefficients are within a range that is dependent on the norm bound ranges.

Further features and advantages of the invention will become more readily apparent from the following detailed description when taken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a system that can be used in practicing embodiments of the invention.

FIG. 2 is a flow diagram of a public key digital signature technique which, when taken with the subsidiary flow diagrams referred to therein, can be used in implementing embodiments of the invention.

FIG. 3 is a flow diagram, in accordance with an embodiment hereof, of a routine for key generation.

FIG. 4 is a flow diagram, in accordance with an embodiment hereof, of a routine for signing and encoding a digital message.

FIG. 5 is a flow diagram, in accordance with an embodiment hereof, of a routine for verification of an encoded digital signature.

DETAILED DESCRIPTION

FIG. 1 is a block diagram of a system that can be used in practicing embodiments of the invention. Two processor-based subsystems 105 and 155 are shown as being in communication over an insecure channel 50, which may be, for example, any wired or wireless communication channel such as a telephone or internet communication channel. The subsystem 105 includes processor 110 and the subsystem 155 includes processor 160. The subsystems can typically comprise mobile devices, computers, or terminals. When programmed in the manner to be described, the processors 110 and 160 and their associated circuits can be used to implement an embodiment of the invention and to practice an embodiment of the method of the invention. The processors 110 and 160 may each be any suitable processor, for example an electronic digital processor or microprocessor. It will be understood that any general purpose or special purpose processor, or other machine or circuitry that can perform the functions described herein, electronically, optically, or by other means, can be utilized. The subsystem 105 will typically include memories 123, clock and timing circuitry 121, input/output functions 118 and display 125, which may all be of conventional types. Inputs can include a touchscreen/keyboard input as represented at 103. Communication is via transceiver 135, which may comprise a modem or any suitable device for communicating signals.

The subsystem 155 in this illustrative embodiment can have a similar configuration to that of subsystem 105. The processor 160 has associated input/output circuitry 164, memories 168, clock and timing circuitry 173, and a display 176. Inputs include a touchscreen/keyboard 155. Communication of subsystem 155 with the outside world is via transceiver 162 which, again, may comprise a modem or any suitable device for communicating signals.

FIG. 2 illustrates a basic procedure that can be utilized with a public key digital signature technique, and refers to routines illustrated by other referenced flow diagrams which describe features in accordance with an embodiment of the invention. Reference can also be made to Appendix I for further details of the invention. The block 210 represents the generating of the public key and private key signals and data, and the publishing of the public key. The routine of an embodiment thereof is described in conjunction with the flow diagram of FIG. 3. In the present example, this operation can be performed, for example, at the processor-based subsystem 105 of FIG. 1. The public key information can be published; that is, made available to any member of the public or to any desired group from whom the private key holder desires to receive the digital signatures. Typically, although not necessarily, the public key may be made available at a central public key library facility or website where a directory of public key holders and their public keys are maintained.

The block 250 represents a routine that can be employed (that is, in this example, by the user of processor-based subsystem 155 of FIG. 1) for signing and encoding the digital message. This routine, in accordance with an embodiment of the invention, is described in conjunction with the flow diagram of FIG. 4. In this example, the encoded digital signature is then transmitted over the channel 50 (FIG. 1).

The block 270 represents a routine that can be employed (that is, in this example, by the user of processor-based subsystem 155 of FIG. 1) for using, inter alia, the public key to implement a verification procedure to either accept or reject the encoded signature. This routine, in accordance with an embodiment of the invention, is described in conjunction with the flow diagram of FIG. 5.

FIG. 3 is a flow diagram of a routine, represented by the block 210 of FIG. 2, in accordance with an embodiment of the invention, for implementing key generation. Reference can also be made to Appendix I. The block 310 represents the inputting of parameters used in key generation, which include: primes N and q, (with N being the dimension for polynomials of degree N−1 and having N ordered coefficients, q=mN+1, and q>>N); Rq, the ring of polynomials Z_(q)[x]/(x^(N)−1); w, a primitive N^(th) root of unity modulo q; {ω} the set of powers of ω, that is, {ω}={ω, ω², ω³, . . . }; Ω a set of t members of (ω), with t approximately N/2; k, an integer, which is a norm bound for the noise; k−b, an integer which is a norm bound for the signatures; and R_(f), the space of private key polynomials, which is a subset of the ring of polynomials, R_(q).

The block 320 represents the random selection of a polynomial f in the space R_(f) of private keys. The polynomial f is the private key. Then, the block 330 represents generation of the public key, F_(Ω)(f). F_(Ω)(f) is obtained by evaluating the polynomial f at the t members of ω in Ω. The block 340 represent the storing of the private key f and the publishing of the public key F_(Ω)(f).

FIG. 4 is a flow diagram of a routine, represented by the block 240 of FIG. 2, in accordance with an embodiment of the invention, for implementing the signing and encoding of a digital message using, inter alia, the private key. Reference can also be made to Appendix I.

Referring to FIG. 4, the block 410 represents the inputting of D, the digital message to be signed, μ, a hash of the digital message D to be signed, and an algorithm called FormatC, which can be used for converting a hash value to a polynomial with small coefficients. (Reference can be made to the above referenced U.S. Pat. No. 6,076,163 with regard to the function implemented by FormatC.) The block 420 represents the random generation of a noise polynomial (also called a commitment polynomial), designated y, with all coefficients having absolute value less than k. Then, the block 430 represents the generation of the hash h, the polynomial c, and the polynomial z. Specifically, the hash h is obtained by applying a hash function to the public key F_(Ω)(y) and μ which is the hash of the digital message D. The algorithm FormatC is then applied to h to obtain c, a polynomial with small coefficients. The polynomial z, which is a candidate digital signature, is then obtained from z=f·c+y.

The decision block 440 represents the step of determining whether the coefficients of the candidate signature are in a predetermined range, dependent on range-defining integers. In this embodiment, a determination is made of whether Norm_(∞)(z) is less than (k−b). If not, the block 420 is re-entered, and the process steps of blocks 420, 430 and 440 are repeated until a candidate digital signature which meets the criterion of block 440 is obtained. The block 450 is then entered, this block representing the outputting of the qualifying candidate signature, that is, the encoded signed message z. Typically the polynomial c, used in obtaining z (or the hash h, from which c can be derived) is also output.

FIG. 5 is a flow diagram of a routine, represented by the block 270 of FIG. 2, in accordance with an embodiment of the invention, for implementing verification of whether the received encoded signed message is valid. Reference can also be made to Appendix I.

The block 510 represents the inputting of parameters that include the encoded signed message (c, z), the hash (p) of the message to be verified and the public key F_(Ω)(f). Typically, the other listed input parameters are also made available; that is: N, q, R_(q) ω, {ω} Ω, k, (k−b), and FormatC, as previously described.

In FIG. 5, as represented by the block 520, the verifier first checks that the encoded signature polynomial has coefficients in the correct range; that is, for this embodiment, determines whether Norm_(∞)(z)<(k−b). If not, the encoded signature is rejected. If so, however, the verification routine continues, using the public key F_(Ω)(f). The polynomial z is evaluated to obtain F_(Ω)(z). Since F_(Ω) is a ring homomorphism, we have that F_(Ω)(z)=F_(Ω)(f) F_(Ω)(c)+F_(Ω)(y), and the verifier can determine F_(Ω)(y) by subtracting the componentwise product of F_(Ω)(f) and F_(Ω)(c) from F_(Ω)(z). The signature is then valid if any only if the hash of F_(Ω)(y) along with p (the hash of the message D) is equal to the received hash value h (or, equivalently, for our purposes, if the short polynomials derived from the respective hashes (e.g. using FormatC) are equal. As represented block 530 of the FIG. 5 embodiment, h′ is the hash of (F_(Ω)(z)−F_(Ω)(f) F_(Ω)(c), μ) where F_(Ω)(z) (resp. c) is the set of evaluations of z (resp. c) at the values of ω in Ω, and c′ is the short polynomial to which h′ is converted using FormatC (that is, c′=FormatC(h′)). Then if c=c′ (block 540), the encoded signature is accepted (block 551). If these quantities are unequal, the signature is rejected (block 552).

The invention has been described with reference to particular preferred embodiments, but variations within the spirit and scope of the invention will occur to those skilled in the art. For example, while a digital signature technique has been described, it will be understood that an authentication producer of the challenge-response-verification type can alternatively be implemented, using the technique hereof by using the challenge as the message to be signed. 

1. A method for signing and subsequently verifying a digital message, comprising the following steps implemented using at least one processor-based subsystem: selecting parameters that include first and second primes, a ring of polynomials related to said primes, and at least one range-defining integer; deriving private and public keys respectively related to a random polynomial private key of the ring of polynomials, and to evaluations of roots of unity of the random polynomial to obtain a public key set of integers; storing the private key and publishing the public key; signing the digital message by: (A) generating a noise polynomial, (B) deriving a candidate signature by obtaining a hash of the digital message and the public key evaluated at the noise polynomial, and determining the candidate signature using the private key, a polynomial derived from the hash, and the noise polynomial, (C) determining whether the coefficients of the candidate signature are in a predetermined range dependent on said at least one range-defining integer, and (D) repeating steps (A) through (C) until the criterion of step (C) is satisfied, and outputting the resultant candidate signature as an encoded signed message; and performing a verification procedure utilizing the encoded signed message and the public key to determine whether the encoded signed message is valid.
 2. The method as defined by claim 1, further comprising transmitting the encoded signed message, and wherein said step of performing a verification procedure includes receiving the transmitted message and performing the verification procedure on the received message.
 3. The message as defined by claim 2, wherein said digital message comprises a challenge communication from a verifier entity, and wherein said encoded signed message is transmitted to said verifier entity.
 4. The method as defined by claim 1, wherein said step of selecting parameters that include at least one range-defining integer comprises selecting parameters that include first and second range-defining integers.
 5. The method as defined by claim 4, wherein said step (C) of said signing of the digital message comprises determining whether the coefficients of the candidate signature are in a predetermined range dependent on said first and second range-defining integers.
 6. The method as defined by claim 4, wherein said first and second range-defining integers define norm bound ranges, and wherein said step of determining whether the coefficients of the candidate signature are in a predetermined range comprises determining whether said coefficients are within a range dependent on the norm bound ranges.
 7. The method as defined by claim 6, wherein said first and second range-defining integers respectively comprise an integer k, which is the infinity-norm of the noise polynomial, and an integer b, which is the 1−norm of the commitment polynomial, and wherein said step of determining whether the coefficients of the candidate signature are in a predetermined range comprises determining whether said coefficients are within the range −(k−b) to (k−b).
 8. A method for signing and transmitting a digital message, comprising the following steps implemented using at least one processor-based subsystem: selecting parameters that include first and second primes, a ring of polynomials related to said primes, and at least one range-defining integer; deriving private and public keys respectively related to a random polynomial private key of the ring of polynomials, and to evaluations of roots of unity of the random polynomial to obtain a public key set of integers; storing the private key and publishing the public key; signing the digital message by: (A) generating a noise polynomial, (B) deriving a candidate signature by obtaining a hash of the digital message and the public key evaluated at the noise polynomial, and determining the candidate signature using the private key, a polynomial derived from the hash, and the noise polynomial, (C) determining whether the coefficients of the candidate signature are in a predetermined range dependent on said at least one range-defining integer, and (D) repeating steps (A) through (C) until the criterion of step (C) is satisfied, and outputting the resultant candidate signature as an encoded signed message; and transmitting the encoded signed message.
 9. The message as defined by claim 8, wherein said digital message comprises a challenge communication from a verifier entity, and wherein said encoded signed message is transmitted to said verifier entity.
 10. The method as defined by claim 8, wherein said step of selecting parameters that include at least one range-defining integer comprises selecting parameters that include first and second range-defining integers.
 11. The method as defined by claim 10, wherein said step (C) of said signing of the digital message comprises determining whether the coefficients of the candidate signature are in a predetermined range dependent on said first and second range-defining integers.
 12. The method as defined by claim 10, wherein said first and second range-defining integers define norm bound ranges, and wherein said step of determining whether the coefficients of the candidate signature are in a predetermined range comprises determining whether said coefficients are within a range dependent on the norm bound ranges.
 13. The method as defined by claim 12, wherein said first and second range-defining integers respectively comprise an integer k, which is the infinity-norm of the noise polynomial, and an integer b, which is the 1−norm of the commitment polynomial, and wherein said step of determining whether the coefficients of the candidate signature are in a predetermined range comprises determining whether said coefficients are within the range −(k−b) to (k−b). 